It's called a beaver because it chews on long pieces of wood, and because of the way it stretches it's mouth to look like the scary movie mask, that to looks like a va jay jay. Probably because it's the part of a woman which has to work so hard to keep mankind on earth, From egesting her superfluous fluids. Her primary reproductive organ. Coupled with the expression, "working as hard as a beaver".
The musings of a feminist medievalist contrarian historian oenophile francophile fromage-fancier. Hint: If you are one of those people who would rather read than make banal chitchat with the hairdresser, take something like this along with you. I promise you will not get asked any more than a single question about it. Really, I would!